**Statistical Physics 2022**

### Statistical Physics (MATH327), Spring Term 2022

Stochastic (probabilistic) processes provide often-outstanding mathematical descriptions of systems within the domain of statistical physics—one of the pillars of modern physics. This module introduces the foundations of statistical physics, including the concepts of statistical ensembles, the laws of thermodynamics, and derived quantities such as entropy. These foundations will be applied to investigate diffusion, the behaviour of idealized physical systems such as classical and quantum gases, thermodynamic cycles, and phase transitions. In particular, numerical computer programming will be used to investigate diffusion and transport in terms of stochastic processes.

### Learning outcomes

Upon completing this module, students are able to:

- Demonstrate understanding of the micro-canonical, canonical and grand-canonical ensembles, their relation and derived concepts such as entropy, temperature and chemical potential.
- Understand the derivation of the equation of state for non-interacting classical and quantum gases.
- Program numerical computer simulations to analyze diffusion from an underlying stochastic process.
- Know the laws of thermodynamics and demonstrate their application to thermodynamic cycles.
- Be aware of the effects of interactions, including an understanding of the origin of phase transitions.

### Background and logistics

This module is intended for third- and fourth-year undergraduates as well as students in the one-year taught MSc programme. Familiarity with combinatorics, quantum mechanics or computer programming is **not** assumed. Any necessary information on these topics will be provided. All resources for the module will be gathered at its Canvas site.

These gapped lecture notes are the main learning resource. There is also a python programming demo illustrating all the tools needed for the computer project. All of these are kept under version control at GitHub, providing an easy way to monitor any typo fixes or other improvements. Reports of any issues, and pull requests to address them, are also welcome.

We will meet at 16:00 on Mondays, 9:00 on Tuesdays & Fridays, and 13:00 on Fridays. I will hold office hours in Room 123 (Theoretical Physics Wing) and online, at 17:00 on Mondays and 10:00 on Fridays, immediately after the corresponding class meetings. You can also make an appointment to meet me at other times.

### How to get the most out of this module

As you know by this point in your studies, the best way to learn mathematics is by doing mathematics. This includes (but is not limited to) making sure you can fill in the gaps in the lecture notes, work through tutorial exercises and solve the homework problems yourself (though I encourage you to discuss these topics with each other). Sample exams are also available, and additional exercises like those discussed in tutorials can be developed by request to offer further opportunities for practice and reinforcement. Should you ask me questions about these exercises and problems, I will endeavour to avoid doing the work for you; instead I will have you explain to me what you have tried so far, and will ask leading questions to suggest where I see problems or potential next steps.

### Additional resources

In addition to the gapped lecture notes, there are tutorial exercises on entropy bounds, Stirling's formula, the mixing entropy, the Otto cycle, the Einstein solid, and the Ising model on various lattice structures, along with a computer project (part one and part two) and homework assignments (one and two).

Potentially useful statistical physics references at roughly the level of this module include:

- David Tong,
*Lectures on Statistical Physics*(2012) - Daniel V. Schroeder,
*An Introduction to Thermal Physics*(2021) - C. Kittel and H. Kroemer,
*Thermal Physics*(1980) - F. Reif,
*Fundamentals of Statistical and Thermal Physics*(1965)

There are also many textbooks at a higher level than this module, including the following:

- R. K. Pathria,
*Statistical Mechanics*(1996) - Sidney Redner,
*A Guide to First-Passage Processes*(2007) - Pavel L. Krapivsky, Sidney Redner and Eli Ben-Naim,
*A Kinetic View of Statistical Physics*(2010) - Kerson Huang,
*Statistical Mechanics*(1987) - Andreas Wipf,
*Statistical Approach to Quantum Field Theory*(2021) - Weinan E, Tiejun Li and Eric Vanden-Eijnden,
*Applied Stochastic Analysis*(2019) - Michael Plischke and Birger Bergersen,
*Equilibrium Statistical Physics*(2005) - L. D. Landau and E. M. Lifshitz,
*Statistical Physics, Part 1*(1980)

Finally, this general book about learning provides useful information about what strategies are most effective, for example retrieval practice compared to re-reading lecture notes or re-watching videos:

- Peter C. Brown, Henry L. Roediger III and Mark A. McDaniel,
*Make it Stick: The Science of Successful Learning*(first edition, 2014). A short summary video is also available

### Schedule

**31 January:** Big-picture overview; Logistics; Probability foundations

**1 February:** Probability foundations; Probability space; Law of large numbers

**4 February:** Law of large numbers; Probability distributions; Central limit theorem; Pseudo-random numbers; Inverse transform sampling

**7 February:** Random walks; Law of diffusion

**8 February:** Diffusion from the central limit theorem; First law of thermodynamics; Micro-canonical ensemble; Thermodynamic equilibrium

**11 February:** Micro-canonical entropy; Second law of thermodynamics; Maximal entropy in thermodynamic equilibrium; Fitting in Python; Spin system entropy bounds

**14 February:** Micro-canonical temperature; Spin system temperature; Heat exchange

**15 February:** Canonical ensemble; Thermal reservoir; Replicas and occupation numbers; Partition function

**18 February:** Boltzmann distribution; Internal energy, heat capacity, and entropy; Helmholtz free energy; Distinguishable vs. indistinguishable spins

**Tutorial:** Spin system entropies and arrow of time; Stirling's formula and bounds on log(N!)

**21 February:** Distinguishable vs. indistinguishable spins—internal energy and entropy depend on information content; Asymptotic limits and approaches

**22 February:** Non-relativistic, classical, ideal gas; Energy levels in a box; Distinguishable vs. indistinguishable canonical partition functions; Internal energy and entropy for indistinguishable case

**25 February:** Internal energy and entropy for ideal gas of distinguishable particles; The mixing entropy and the 'Gibbs paradox'

**Tutorial:** Stirling's formula, series expansion and applications

**28 February:** Pressure, ideal gas law, equations of state; Work, pressure and force

**1 March:** Heat and entropy; First law of thermodynamics; PV diagrams, isotherms and adiabats

**4 March:** PV diagrams, isotherms and adiabats; Carnot cycle; Heat engine efficiency

**Tutorial:** Maxwell's demon; Mixing entropy

**7 March:** Carnot cycle efficiency; Refrigerators; Grand-canonical ensemble

**8 March:** Grand-canonical ensemble; Chemical potential; Particle reservoir; Grand-canonical partition function

**11 March:** Grand-canonical partition function; Grand-canonical potential; Generalized thermodynamic identity

**Tutorial:** Cauchy–Lorentz distribution; Generalized diffusion length; Anomalous diffusion; Mixing entropy; Otto cycle

**14 March:** Maxwell–Boltzmann statistics; Quantum statistics; Quantized energy levels and their micro-states

**15 March:** Bosons vs. fermions; Bose–Einstein statistics; Factorization; Fermi–Dirac statistics; The classical limit

**18 March:** Average occupation numbers in the high-temperature classical limit; Non-relativistic ideal gas of bosons

**Tutorial:** Anomalous diffusion in biophysics; ST diagrams; Otto cycle efficiency; Diesel cycle

**21 March:** Ideal boson gases; Ultra-relativistic photon energies, wavelengths and frequencies; Photon gas internal energy density and Planck spectrum

**22 March:** Planck spectrum vs. Rayleigh–Jeans ultraviolet catastrophe; Solar radiation and cosmic microwave background; Radiation pressure

**25 March:** Photon gas equation of state; Non-relativistic quantum fermion gases at low temperature; Fermi function

**Tutorial:** Photon polarizations; Dark matter in cosmic microwave background angular power spectrum; Heat capacity of materials; Einstein solid

**28 March:** Low-temperature Fermi function; Fermi energy; Constant energy and pressure (equation of state) for non-relativistic fermion gas at low temperature

**29 March:** Degeneracy pressure; Type-Ia supernovas; Ultra-relativistic neutrino gas equation of state

**1 April:** Density of states; Sommerfeld expansion for low-temperature fermion gas: chemical potential and heat capacity

**Tutorial:** Einstein solid; Phonons; Debye theory of solids

**25 April:** Logistics and post-break recap; Phase transitions; Interacting systems

**26 April:** Ising model; Magnetization; Ordered and disordered phases

**29 April:** Order parameters; Phase transitions; Magnetization as order parameter; Ising model mean-field approximation

**Tutorial:** Ising model on different lattice structures; Coordination number; Frustrated anti-ferromagnet; Spin glasses; Fully connected lattice

**3 May:** Mean-field self-consistency condition, critical temperature and critical exponent; Mean-field reliability in various dimensions

**6 May:** Mean-field reliability in various dimensions; Ising model exact solution in one dimension

**Tutorial:** Frustrated anti-ferromagnet; Ising model on fully connected lattice (Curie–Weiss model) exact solution

**9 May:** Kramers–Wannier duality for Ising model in two dimensions; Link variables; Dual lattice

**10 May:** Exact critical temperature from Kramers–Wannier duality; Numerical methods; Monte Carlo integration

**13 May:** Monte Carlo importance sampling; Metropolis--Rosenbluth--Teller algorithm; Universality; Broader applications; Module recap

Last modified 13 May 2022